Lie Symmetries and Exact Solutions of the Generalized Thin Film Equation

TL;DR

This paper classifies symmetries of generalized fourth-order reaction-diffusion equations, constructs exact solutions including self-similar and traveling wave solutions, and extends previous Lie symmetry analyses to more general cases.

Contribution

It generalizes Lie symmetry classification for complex fourth-order reaction-diffusion equations and constructs new exact solutions using symmetry reductions.

Findings

Identified symmetry groups for generalized reaction-diffusion equations.

Constructed exact solutions including self-similar and traveling waves.

Extended previous symmetry analysis to more general equations.

Abstract

A symmetry group classification for fourth-order reaction-diffusion equations, allowing for both second-order and fourth-order diffusion terms, is carried out. The fourth order equations are treated, firstly, as systems of second-order equations that bear some resemblance to systems of coupled reaction-diffusion equations with cross diffusion, secondly, as systems of a second-order equation and two first-order equations. The paper generalizes the results of Lie symmetry analysis derived earlier for particular cases of these equations. Various exact solutions are constructed using Lie symmetry reductions of the reaction-diffusion systems to ordinary differential equations. The solutions include some unusual structures as well as the familiar types that regularly occur in symmetry reductions, namely self-similar solutions, decelerating and decaying traveling waves, and steady states.